| L(s) = 1 | − 0.0270·2-s + 3-s − 1.99·4-s − 4.41·5-s − 0.0270·6-s + 1.75·7-s + 0.107·8-s + 9-s + 0.119·10-s + 5.53·11-s − 1.99·12-s − 13-s − 0.0474·14-s − 4.41·15-s + 3.99·16-s − 6.56·17-s − 0.0270·18-s − 6.47·19-s + 8.82·20-s + 1.75·21-s − 0.149·22-s + 3.83·23-s + 0.107·24-s + 14.4·25-s + 0.0270·26-s + 27-s − 3.51·28-s + ⋯ |
| L(s) = 1 | − 0.0190·2-s + 0.577·3-s − 0.999·4-s − 1.97·5-s − 0.0110·6-s + 0.664·7-s + 0.0381·8-s + 0.333·9-s + 0.0376·10-s + 1.66·11-s − 0.577·12-s − 0.277·13-s − 0.0126·14-s − 1.13·15-s + 0.998·16-s − 1.59·17-s − 0.00636·18-s − 1.48·19-s + 1.97·20-s + 0.383·21-s − 0.0318·22-s + 0.798·23-s + 0.0220·24-s + 2.89·25-s + 0.00529·26-s + 0.192·27-s − 0.664·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.019292352\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.019292352\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 + 0.0270T + 2T^{2} \) |
| 5 | \( 1 + 4.41T + 5T^{2} \) |
| 7 | \( 1 - 1.75T + 7T^{2} \) |
| 11 | \( 1 - 5.53T + 11T^{2} \) |
| 17 | \( 1 + 6.56T + 17T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 23 | \( 1 - 3.83T + 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 - 3.17T + 31T^{2} \) |
| 37 | \( 1 - 1.56T + 37T^{2} \) |
| 41 | \( 1 + 5.06T + 41T^{2} \) |
| 43 | \( 1 + 2.34T + 43T^{2} \) |
| 47 | \( 1 + 3.38T + 47T^{2} \) |
| 53 | \( 1 - 2.91T + 53T^{2} \) |
| 59 | \( 1 - 4.28T + 59T^{2} \) |
| 61 | \( 1 + 0.610T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 + 3.98T + 73T^{2} \) |
| 79 | \( 1 - 9.21T + 79T^{2} \) |
| 83 | \( 1 + 0.944T + 83T^{2} \) |
| 89 | \( 1 - 2.67T + 89T^{2} \) |
| 97 | \( 1 + 7.30T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.413345223718190497101155092873, −8.029246027340366450405501052492, −7.04535616528731918643546798989, −6.58996332251568953203992502313, −5.03023908775494917352303973090, −4.35416167676346640857743724819, −4.02494902557386301993587725371, −3.35598713513292187092921056746, −1.89253598049138860054921528093, −0.57055515521002132195335096437,
0.57055515521002132195335096437, 1.89253598049138860054921528093, 3.35598713513292187092921056746, 4.02494902557386301993587725371, 4.35416167676346640857743724819, 5.03023908775494917352303973090, 6.58996332251568953203992502313, 7.04535616528731918643546798989, 8.029246027340366450405501052492, 8.413345223718190497101155092873
